Found problems: 4776
Replacing any of the coefficients of quadratic trinomial $f(x)=ax^2+bx+c$ with an $1$ will result in a quadratic trinomial with at least one real root. Prove that the resulting trinomial attains a negative value at at least one point.
EDIT: Oops I failed, added "with a 1." Also, I am sorry for not knowing these are posted already, however, these weren't posted in the contest lab yet, which made me think they weren't translated yet.
Note: fresh translation
Determine all positive integers $n$ such that the number $n(n+2)(n+4)$ has at most $15$ positive divisors.
The game of $pebbles$ is played on an infinite board of lattice points $(i,j)$. Initially there is a $pebble$ at $(0,0)$. A move consists of removing a $pebble$ from point $(i,j)$and placing a $pebble$ at each of the points $(i+1,j)$ and $(i,j+1)$ provided both are vacant. Show taht at any stage of the game there is a $pebble$ at some lattice point $(a,b)$ with $0 \leq a+b \leq 3$
Let $ K$ be the figure bounded by the graph of function $ y \equal{} \frac {x}{\sqrt {1 \minus{} x^2}}$, $ x$ axis and the line $ x \equal{} \frac {1}{2}$.
(1) Find the volume $ V_1$ of the solid generated by rotation of $ K$ around $ x$ axis.
(2) Find the volume $ V_2$ of the solid generated by rotation of $ K$ around $ y$ axis.
Please solve question (2) without using the shell method for Japanese High School Students those who don't learn it.
A round-robin tournament among $2n$ teams lasted for $2n-1$ days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the $n$ games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one winning team from each day without choosing any team more than once?
Prove that there is a function $f$ from the set of all natural numbers into itself such that $f(f(n))=n^2$ for all $n \in \mathbb{N}$.
Let $S$ be the set of all polygonal areas in a plane. Prove that there is a function $f : S \to (0,1)$ which satisfies $f(S_1 \cup S_2) = f(S_1)+ f(S_2)$ for any $S_1,S_2 \in S$ which have common points only on their borders
Let $f:\mathbb N\to\mathbb R$ be given by
$$f(n)=n^{\frac12\tau(n)}$$for $n\in\mathbb N=\{1,2,\ldots\}$ where $\tau(n)$ is the number of divisors of $n$. Show that $f$ is an injection.
Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$.
$\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$
For any real $ x$, let $ \lfloor x\rfloor$ be the largest integer that is not more than $ x$. Given a sequence of positive integers $ a_1,a_2,a_3,\ldots$ such that $ a_1>1$ and
\[ \left\lfloor\frac{a_1\plus{}1}{a_2}\right\rfloor\equal{}\left\lfloor\frac{a_2\plus{}1}{a_3}\right\rfloor\equal{}\left\lfloor\frac{a_3\plus{}1}{a_4}\right\rfloor\equal{}\cdots\]
Prove that
\[ \left\lfloor\frac{a_n\plus{}1}{a_{n\plus{}1}}\right\rfloor\leq1\]
holds for every positive integer $ n$.
Find the sum of all positive rational numbers that are less than $10$ and that have denominator $30$ when written in lowest terms.
If $f$ is a function from the set of positive integers to itself such that $f(x) \leq x^2$ for all natural $x$, and $f\left( f(f(x)) f(f(y))\right) = xy$ for all naturals $x$ and $y$. Find the number of possible values of $f(30)$.
[i]Author: Alex Zhu[/i]
Show that there are no integers $a,b,c$ for which $a^2+b^2-8c=6$.
Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.
Let $ f(x)\equal{}\left(1\plus{}\frac{1}{x}\right)^{x}\ (x>0)$.
Find $ \lim_{n\to\infty}\left\{f\left(\frac{1}{n}\right)f\left(\frac{2}{n}\right)f\left(\frac{3}{n}\right)\cdots\cdots f\left(\frac{n}{n}\right)\right\}^{\frac{1}{n}}$.
Given a function $f\colon \mathbb{R}\rightarrow \mathbb{R} $ with $f(x)^2\le f(y)$ for all $x,y\in\mathbb{R} $, $x>y$, prove that $f(x)\in [0,1] $ for all $x\in \mathbb{R}$.
[b]Problem 3.[/b] Let $n\geq 3$ is given natural number, and $M$ is the set of the first $n$ primes. For any nonempty subset $X$ of $M$ with $P(X)$ denote the product of its elements. Let $N$ be a set of the kind $\ds\frac{P(A)}{P(B)}$, $A\subset M, B\subset M, A\cap B=\emptyset$ such that the product of any 7 elements of $N$ is integer. What is the maximal number of elements of $N$?
[i]Alexandar Ivanov[/i]
(a) Find all strictly monotone functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(x+f(y))=f(x)+y\quad\text{for all real}\ x,y. \]
(b) If $n>1$ is an integer, prove that there is no strictly monotone function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[ f(x+f(y))=f(x)+y^n\quad \text{for all real}\ x, y.\]
Let $ q_{0}, q_{1}, \cdots$ be a sequence of integers such that
a) for any $ m > n$, $ m \minus{} n$ is a factor of $ q_{m} \minus{} q_{n}$,
b) item $ |q_n| \le n^{10}$ for all integers $ n \ge 0$.
Show that there exists a polynomial $ Q(x)$ satisfying $ q_{n} \equal{} Q(n)$ for all $ n$.
Let $p$ a positive prime number and $\mathbb{F}_{p}$ the set of integers $mod \ p$. For $x\in \mathbb{F}_{p}$, define $|x|$ as the cyclic distance of $x$ to $0$, that is, if we represent $x$ as an integer between $0$ and $p-1$, $|x|=x$ if $x<\frac{p}{2}$, and $|x|=p-x$ if $x>\frac{p}{2}$ . Let $f: \mathbb{F}_{p} \rightarrow \mathbb{F}_{p}$ a function such that for every $x,y \in \mathbb{F}_{p}$
\[ |f(x+y)-f(x)-f(y)|<100 \]
Prove that exist $m \in \mathbb{F}_{p}$ such that for every $x \in \mathbb{F}_{p}$
\[ |f(x)-mx|<1000 \]
Problem 3.
Let $f:\left[ 0,\frac{\pi }{2} \right]\to \left[ 0,\infty \right)$ an increasing function .Prove that:
(a) $\int_{0}^{\frac{\pi }{2}}{\left( f\left( x \right)-f\left( \frac{\pi }{4} \right) \right)}\left( \sin x-\cos x \right)dx\ge 0.$
(b) Exist $a\in \left[ \frac{\pi }{4},\frac{\pi }{2} \right]$ such that $\int_{0}^{a}{f\left( x \right)\sin x\ dx=}\int_{0}^{a}{f\left( x \right)\cos x\ dx}.$
You have to organize a fair procedure to randomly select someone from $ n$ people so that every one of them would be chosen with the probability $ \frac{1}{n}$. You are allowed to choose two real numbers $ 0<p_1<1$ and $ 0<p_2<1$ and order two coins which satisfy the following requirement: the probability of tossing "heads" on the first coin $ p_1$ and the probability of tossing "heads" on the second coin is $ p_2$. Before starting the procedure, you are supposed to announce an upper bound on the total number of times that the two coins are going to be flipped altogether. Describe a procedure that achieves this goal under the given conditions.
If the equation:
$f(\frac{x-3}{x+1}) + f(\frac{3+x}{1-x}) = x$
holds true for all real x but $\pm 1$, find $f(x)$.
Find all functions $f,g: \mathbb{Q}\to \mathbb{Q}$ such that the following two conditions hold:
$$f(g(x)-g(y))=f(g(x))-y \ \ (1)$$
$$g(f(x)-f(y))=g(f(x))-y\ \ (2)$$
for all $x,y \in \mathbb{Q}$.
(1) Let $f(x)$ be the differentiable and increasing function such that $f(0)=0.$Prove that $\int_{0}^{1}f(x)f'(x)dx\geq \frac{1}{2}\left(\int_{0}^{1}f(x)dx\right)^{2}.$
(2) $g_{n}(x)=x^{2n+1}+a_{n}x+b_{n}\ (n=1,\ 2,\ 3,\ \cdots)$ satisfies $\int_{-1}^{1}(px+q)g_{n}(x)dx=0$ for all linear equations $px+q.$
Find $a_{n},\ b_{n}.$